HIGHLY EFFICIENT SPECTRAL CALIBRATION METHODS FOR

SWEPT-SOURCE OPTICAL COHERENCE TOMOGRAPHY

A Thesis

by

OSCAR JOSEU PACHECO BARAJAS

Submitted to the Office of Graduate and Professional Studies ofTexas A&M University

in partial fulfillment of the requirements for the degree of

MASTER OF SCIENCE

Chair of Committee, Sebastian HoyosCommittee Members, Javier A. Jo

Kamran EntesariRaffaella Righetti

Head of Department, Miroslav M. Begovic

August 2017

Major Subject: Electrical Engineering

Copyright 2017 Oscar Joseu Pacheco Barajas

ABSTRACT

Recent techniques in optical coherence tomography (OCT) make use of specialized

light sources that sweep across a broad optical bandwidth, allowing for longer depth

ranges at higher resolutions.The produced light source signal can be described as a gaus-

sian damped sinusoid that non-uniformly sweeps across a narrow frequency band. When

sampling this interferometric signal uniformly, the generated images present considerable

distortion, because the spectral information is a function of wavenumber "k", not time. To

solve this problem a "calibration" step needs to be performed; in this process, the acquired

interferogram signal is linearized into k-space. The process usually involves estimating the

phase-frequency change profile of the SS-OCT system via Hilbert transformation, inverse

tangent and phase unwrapping.

In this thesis, a multitude of low complexity, computationally efficient methods for

the real-time calibration of Swept Source Optical Coherence Tomography (SS-OCT) sys-

tems are implemented and results are evaluated against commonly performed calibration

techniques such as Hilbert transformation. Simulation shows execution times decisively

improved by up to a factor of ten, depending on the used technique. Axial resolution was

also slightly improved across all the tested techniques. Moreover, the inverse tangent and

phase unwrapping steps necessary for Hilbert transform calibration techniques are elimi-

nated, vastly reducing circuit implementation complexity and making the system suitable

for future inexpensive, power efficient, on-chip solutions in SS-OCT post-processing.

ii

DEDICATION

To my mother, for it is her love, encouragement, and constant support that has guided me

along the way.

iii

ACKNOWLEDGMENTS

I would like to thank all those who contributed in some manner to the work described

in this thesis. First and foremost I would like to thank my graduate advisor, professor Se-

bastian Hoyos, for his constant support and for providing me with intellectual freedom. It

was through his guidance that this thesis came to be at all, his appreciation for innovative

ideas and his support for my attendance to multiple conferences contributed undoubtedly

to generate the core ideas for this work. I’d also like to thank my lab mate, Mr. Amir

Tofighi, for his friendship and meaningful conversations and for his help obtaining the

results and implementing the algorithms shown in this work. lastly, I would like to ac-

knowledge the Mexican council of science and technology (CONACYT) for financially

supporting my graduate studies.

iv

CONTRIBUTORS AND FUNDING SOURCES

Contributors

This work was supported by a thesis committee consisting of Professor Sebastian

Hoyos [advisor], Professor Raffaella Righetti, and Professor Kamran Entesari of the De-

partment of Electrical Engineering, and by Professor Javier Jo of the Department of Biomed-

ical engineering.

The data analyzed for this work was provided by Mr. Michael Serafino of the depart-

ment of Biomedical Engineering.

All other work conducted for the thesis (or) dissertation was completed by the student

independently.

Funding Sources

Graduate study was supported by a scholarship from the Mexican council of science

and technology (CONACYT).

v

NOMENCLATURE

OCT Optical Coherence Tomography

OLCR Optical Low Coherence Reflectometry

OFDI Optical Frequency-domain Imaging

MZI Mach-Zehnder Interferometer

SS-OCT Swept-Source Optical Coherence Tomography

TD-OCT Time-Domain Optical Coherence Tomography

FD-OCT Fourier Domain Optical Coherence Tomography

SD-OCT Spectral-Domain Optical Coherence Tomography

FPGA Field-Programmable Gate Array

CPU Central Processing Unit

ADC Analog to Digital Converter

DAC Digital to Analog Converter

VCSEL Swept vertical cavity surface-emitting lasers

FDML Fourier domain mode locked laser

SLED Superluminescence diode

TSLED Tunable SLED

SNR Signal to noise ratio

SOA Semiconductor Optical Amplifier

MEMS Micro-electro-mechanical systems

RIN Relative Intensity Noise

FWHM Full-Width Half Maximum

vi

PSF Point-spread Function

DFT Discrete Fourier Transform

IpDFT Interpolated discrete fourier transform

FFT Fast Fourier Transform

FFP-TF Fiber Fabry-Perot tunable filters

VT-DBR Vernier-tuned distributed Bragg reflector

DT-SS Dispersion Tuning Swept Sources

ASE Amplified spontaneous emission

vii

TABLE OF CONTENTS

Page

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

DEDICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

CONTRIBUTORS AND FUNDING SOURCES . . . . . . . . . . . . . . . . . . v

NOMENCLATURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

TABLE OF CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

1. INTRODUCTION AND THEORETICAL BACKGROUND . . . . . . . . . . 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Optical Coherence Tomography . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Operating Principles . . . . . . . . . . . . . . . . . . . . . . . . 31.2.2 OCT Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.3 TD-OCT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2.4 FD-OCT or Optical Frequency-domain Imaging . . . . . . . . . . 71.2.5 SD-OCT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.2.6 SS-OCT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.2.7 The Need For Multi-MHZ SS-OCT . . . . . . . . . . . . . . . . 10

2. THE "CALIBRATION" PROBLEM . . . . . . . . . . . . . . . . . . . . . . . 11

2.1 Common Calibration Techniques . . . . . . . . . . . . . . . . . . . . . . 122.1.1 Spectral Phase-based Techniques . . . . . . . . . . . . . . . . . . 122.1.2 Generic Real-time Calibration techniques . . . . . . . . . . . . . 13

3. CALIBRATION BASED ON A SQUARE-LAW ENVELOPE DETECTOR . . 14

3.1 Amplitude Correction by Means of Square-law Envelope Detection . . . . 143.2 Continuous Time Ternary Encoding (CTTE) . . . . . . . . . . . . . . . . 16

viii

3.2.1 CTTE Threshold Determination . . . . . . . . . . . . . . . . . . 163.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4. KALMAN FILTERING BASED CALIBRATION . . . . . . . . . . . . . . . 22

4.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.1.1 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.1.1.1 State equations . . . . . . . . . . . . . . . . . . . . . . 244.1.1.2 Observation equations . . . . . . . . . . . . . . . . . . 26

4.1.2 CLK Generating Mechanism . . . . . . . . . . . . . . . . . . . . 274.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5. INTERPOLATED DISCRETE FOURIER TRANSFORM BASED CALIBRA-TION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.2 IpDFT Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.2.1 Time Windows . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.2.1.1 Background Theory . . . . . . . . . . . . . . . . . . . 335.2.1.2 Betrocco-Yoshida (BY-2) algorithm . . . . . . . . . . . 355.2.1.3 RVCI windows (Order M) . . . . . . . . . . . . . . . . 355.2.1.4 Clock generation . . . . . . . . . . . . . . . . . . . . . 36

5.2.2 Algorithmic Complexity Comparison . . . . . . . . . . . . . . . 375.2.2.1 Core Operation Time/Arithmetic Complexity . . . . . . 375.2.2.2 Auxiliary Operations Time/Arithmetic Complexity . . . 38

5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

6. SUMMARY AND CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . 41

6.1 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416.2 Further Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

ix

LIST OF FIGURES

FIGURE Page

1.1 Michelson interferometer-based OCT system. A reference mirror and abeamsplitter are used to produce an interference signal at the detector. . . 3

1.2 Time Domain, Michelson interferometer-based OCT system. A referencemirror is mechanically moved across an axis to resolve the depth depen-dant coefficients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Spectral-domain based OCT system. A reference mirror remains fixed, asa broadband light source is used to resolve the depth dependant coefficients. 9

1.4 Swept-source OCT system. A reference mirror remains fixed, as a "swept"optical source is used to resolve the depth dependant coefficients. . . . . . 9

2.1 Instantaneous phase linearization (a): unwrapped phase of the MZI (b):MZI signal representation. Dots depict sampling indexes. . . . . . . . . . 12

2.2 Simple zero crossing detector (left) (R1 R2). Clock signal generated atzero-crossings (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.1 Envelope detection process from a continuous time perspective. Figures(c) to (f) correspond to points I-IV in (b), respectively. . . . . . . . . . . . 16

3.2 Continuous time level crossing circuit implementation (top): continuoustime ternary encoding (CTTE) (bottom): K-linear clock generation exam-ple, with N=8 CTTE levels . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.3 Axial resolution mean value (a) to (c) different sampling arm lengths (a:514µm-b:998µm-c:1483µm), across 500 realizations as a function of ADC reso-lution (8bits-14bits) and N (d): sensitivity roll-off across all depths. . . . . 19

3.4 Image reconstruction comparison. (a) Uncalibrated. (b) through (d), in-creasing number of angular quantization levels N. . . . . . . . . . . . . . 20

4.1 SS-OCT optic elements configuration. PC: polarization controller, ODL:optic delay line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

x

4.2 Estimation technique block diagram. σ2w is the measurement noise power.

σ2nA

is the amplitude noise power. σ2nφ

is the instantaneous phase noise power 23

4.3 MZI signal parameter estimation for a typical realization. (a) Amplitudeparameter extraction. (b) Instantaneous phase extraction . . . . . . . . . . 25

4.4 Comparison step circuit. (Top left) Equally divided calibration profile. ∆defines the number of points designed to be sampled (wave number quan-tization interval). (Bottom Left) Calibrating CLK. (Right) Asynchronoustime comparison block diagram. . . . . . . . . . . . . . . . . . . . . . . 26

4.5 Image reconstruction comparison. (a) Uncalibrated. (b) through (d), in-creasing number of wavenumber quantization levels P. . . . . . . . . . . . 28

4.6 Axial Resolution mean value (a) to (c) Different sampling arm lengths(a:1448µm-b:1921µm-c:2361µm), across 500 realizations as a function ofADC Resolution (8bits-14bits) and P (d): sensitivity roll-off (20log10, dB)across all depths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5.1 IpDFT algorithm implementation diagram, bottom right shows a split-radix FFT butterfly diagram. . . . . . . . . . . . . . . . . . . . . . . . . 32

5.2 (a) Normalized execution time compared to Hilbert transformation vs. Ob-servation window length, (b) Axial resolution compared to sample armdepth, (c) Axial resolution vs. Observation window length. . . . . . . . . 38

5.3 Image reconstruction comparison. (a) Uncalibrated. (b) Bertocco-Yoshida(BY-2) (c) RVCI (M=1) (d) RVCI (M=3) . . . . . . . . . . . . . . . . . . 40

xi

LIST OF TABLES

TABLE Page

3.1 Comparison between the axial resolution of the proposed method and thetheoretical spectral phase-based technique. Proposed Method (Top). Spec-tral phase-based (Bottom). All quantities are in µm . . . . . . . . . . . . 20

5.1 Computational complexity comparison of a Hilbert transformation basedtechnique against IpDFT utilizing multiple windows and enhanced splitradix FFT. Signal length is defined as N. . . . . . . . . . . . . . . . . . . 33

5.2 Multiplication algorithms complexity, log∗ stands for iterated logarithm. . 34

5.3 Comparison between the complexity of auxiliary blocks needed for Hilberttransformation techniques and IpDFT. M(N) below stands in for the com-plexity of the chosen multiplication algorithm. . . . . . . . . . . . . . . . 34

xii

1. INTRODUCTION AND THEORETICAL BACKGROUND

This section does not aim to provide the reader with a comprehensive introduction

to the vast fields of Optical coherence tomography (OCT) and Optical frequency-domain

imaging (OFDI), but rather with a succinct summary of what I consider to be the most

critical concepts necessary to understand the present work. However, one could easily

utilize the works referenced in this thesis as a starting point, if a full literary review is of

interest.

In the following sub-sections, the fundamental operating principles of the discussed

imaging technique and others are discussed, together with some information on the dif-

ferent optical light source technology employed. This is done in an effort to elucidate the

motivation behind all the employed calibration techniques and their potential applicability

to both commercial and research/academic environments.

1.1 Motivation

Current SS-OCT systems perform most of the signal processing required to obtain high

resolution brightness maps and 3D images "offline"; meaning they make use of external

processing units such as personal computers running third party software, however, the

use of full personal computers and licensed software can both add to the overall cost of

the system, and ameliorate its efficiency. This creates an opportunity for signal process-

ing, embedded and mixed-signal oriented people to develop, test and implement on-chip

solutions that would perform the necessary operations on a printed circuit board, improv-

ing cost, complexity and eventually leading to the development of portable, point-of-care

devices that can positively affect the adoption of the technology, ultimately improving the

accurate diagnosis of a number of pathologies.

1

1.2 Optical Coherence Tomography

Optical Coherence Tomography (OCT) is a noninvasive imaging technique capable of

generating cross-sectional images of tissue architectural morphology with high axial reso-

lution [1]. Since its introduction as a non-invasive crosssectional imaging technique with a

spatial resolution of a few micrometers back in 1991 [2], Optical Coherence Tomography

(OCT) has been actively developed and commercialized to the point of becoming a highly

efficient medical imaging tool for the accurate diagnosis of a wide variety of pathologies,

especially those of interest in the area of cardiology, optometry, and ophthalmology. Re-

cently, Optical Coherence Tomography has established itself as the de-facto biomedical

imaging modality when low penetration and high resolution are desired. The technology

has made its way into a wide array of applications and fields ranging from biomedical

imaging to archeology. This rapid adoption growth can be attributed at least in part to the

techniques ability to capture high-resolution cross-sectional images without touching the

eye [18]. In the most basic OCT apparatus, back scattered photons from a tissue of interest

are detected using a Michelson interferometer; depth information is then extracted, from

which an image is reconstructed. Subsequently, more advanced techniques that exploit this

basic principle of operation have emerged. Early embodiments of the technique existed

as a time domain interferometric technique in which each depth-dependant reflection, cor-

responds to a different time delay. Each delay is then measured by manually moving the

reference arm of the Michelson interferometer so that depth information can be obtained at

distinct times, making the information time encoded; hence the time domain OCT or TD-

OCT name.[19] Later, the need for faster acquisition speeds has pushed towards more ele-

gant techniques in which the reflection coefficient information is obtained via the Fourier

transform in the wavenumber space (k-space), these techniques are generally refered to as

Fourier domain OCT (FD-OCT).

2

1.2.1 Operating Principles

Optical coherence tomography is fundamentally based on the one dimensional tech-

nique called "optical low coherence reflectometry" (OLCR) [33] which employs a Michel-

son interferometer to detect back reflected light from a sample arm. A Michaelson interfer-

ometer works by splitting from a low coherence source into two directions, usually referred

to as the reference arm and the sample arm. In an optical fiber based implementation of a

Michaelson interferometer, light in the reference arm is directed back after reflection from

a mirror to a 2x2 fiber coupler, where it gets combined with the backscattered light from

the sample in the sample arm in order to generate an interference pattern that contains

information about the depth reflectivity profile of the sample.

Figure 1.1: Michelson interferometer-based OCT system. A reference mirror and a beam-splitter are used to produce an interference signal at the detector.

3

To understand the basics of image formation in an OCT system, consider the sim-

plest case in which the sample is modeled as a collection of N discrete reflectors having

electric field reflectivities rs1, rs2, · · · , rsN and located at the distances zs1, zs2, · · · , zsN

from the beamsplitter as shown in Figure 1.1. The depth-dependant reflectivity profile is

then rs(zs), such a sample can then be mathematically represented as a sum of Dirac-delta

pulses by: rs(zs) =∑N

n=1 rsnδ(zs − zsn) by definition the power reflectivity is equal to

the magnitude squared of its electric ïnAeld reflectivity, that is given by Rsn = |Rsn|2.

Diferentiation is then obtained by reconstructing rs(zs) from the measurements. The light

source electric field can be expressed as Ei = s(k, ω)ei(kz−ωt), where s(k, ω) represents

monochromatic light contribution to the electric field amplitude of the polychromatic light

with a wavenumber k = (2π/λ) and an angular frequency ω = 2πν. As usual, the wave-

length λ and frequency ν are related by the refractive index n(λ) of the medium of propa-

gation asn(λ) = c/νλ , where c denotes the velocity of light in vacuum. The electric field

of the light passing through the 50/50 beam-splitter after getting reflected from the sample

can then be expressed as Es = Ei/√

2(rs ∗ ei2kzs), where * denotes convolution, the 1/√

2

represents 50% reduction in light intensity due to the beam-splitter, lastly, the factor of 2

in the exponential term represent the round-trip path of the light in the sample arm. The

expression for Es can be simplified to Es = Ei/(√

2(rs ∗ei2kzsn), Similarly the expression

for the reference arm electric field can be expressed as ER = Ei/(√

2(rR ∗ ei2kzR) The in-

terference signal detected at the photo-detectorID(K) is the time average of the intensity

signal resulting from the phasor sum of ER and ES expressed by:

ID(k, ω) = (ρ/2)〈|ER + ES|〉 (1.1)

where ρ denotes the responsivity of the photodetector and the factor of 2 reflects the halv-

ing of intensity on the second pass of light through the beam-splitter. Simplifying the

4

above expression based on the fact that the electric field oscillations are much faster than

the responsivity of the photodetector, we get an expression involving only time invariant

terms as:

ID(k) =ρ

4[S(k)(RR +RS1 + · · ·+RSN)]+ (1.2)

1.2.2 OCT Techniques

Over the past two decades, multiple techniques involving OCT have been introduced.

These can be categorized into three main generations based on performance, reliability,

and implementation. First generation techniques relied on the physical movement of the

fiber or optical path for tissue penetration and image reconstruction. This mechanical

time-varying movement led to it’s reference as time domain OCT (TD-OCT), which uses

a single-element photodetector to acquire the fringe signal serially. This technique suffers

from limitations such as a slow scan speed and lower axial resolution; the second and

third generations of OCT techniques can be further categorized as Fourier Domain OCT

(FD-OCT).

In FD-OCT techniques, the spectral information of the interferogram is measured and

the reflection coefficient information is obtained via the Fourier transform in the wavenum-

ber space (k-space) to generate a depth dependant brightness map, commonly known in

tomography as B-scan. These techniques are very similar in concept as no mechanical

movement of the interferometer arm is involved, but they differ in execution. The first

of these techniques is referred to as spectral domain OCT or SD-OCT for short; in this

technique information is obtained by sequentially measuring different wavelengths gener-

ated from a broadband optical source using a spectrometer and a multi-channel analyzer.

While the axial resolution is improved from TD-OCT techniques, the acquisition speed

is limited by the spectrometer and analyzer, also the use of high speed cameras as detec-

5

tion devices can be quite costly, especially in the 1060nm to 1310nm wavelengths [12].

The third generation OCT technique is known as swept source optical coherence tomog-

raphy (SS-OCT). In these technique spectral discrimination is achieved by rapidly tuning

a narrowband light source over a broad optical bandwidth. Since all the light is collapsed

into a single wavelength that is consequently swept across a wide range, better signal to

noise ratios, faster speeds, a better sensitivity roll-off, and tissue penetration compared to

SD-OCT is achieved [13].

Swept Source Optical Coherence Tomography (SS-OCT) systems employ a short-

cavity laser source that sweeps across a narrow band of wavelengths with each scan, these

lasers have a wavelength typically centered at approximately 1 Âtm and can achieve very

high imaging speeds up to 150,000 per second [34] allowing for fast and accurate diagno-

sis; this is particularly important in areas such as angiography where the patient is asked

to hold still for three to four seconds in order to retrieve a 3mm by 3mm image, larger

images would prove nearly imposible as it is very difficult for a patients eye movement to

remain still for longer periods of time. Moreover SS-OCT exhibits higher axial resolution

and an improved signal-to-noise ratio compared to SD-OCT [2], additionally, swept light

sources are invisible, making them more suitable for a diagnostics environment as they

are less distractive. Finally, the longer wavelength exhibited by swept sources provides a

better sensitivity roll-off over the entire scan window, which delivers the ability to obtain

clear images of deeper tissue.

In the following subsections a brief review of the main Optical Coherence Tomography

techniques is presented, starting with time-encoded OCT and following with the two most

common FD-OCT techniques.

6

1.2.3 TD-OCT

In this iteration of the technology, the reference arm of a Michelson’s interferometer

is mechanically moved along an axis, causing the information to be time-encoded, hence

the Time-Domain name for the technique. This mechanical movement causes it to suffer

from lower ranging accuracy and slow tinning speed when compared to modern Fourier

transform based techniques that utilize a less complex setup and no mechanical move-

ment of the reference arm, and can potentially achieve much higher axial resolutions. The

following figure represents a common TD-OCT setup.

Figure 1.2: Time Domain, Michelson interferometer-based OCT system. A referencemirror is mechanically moved across an axis to resolve the depth dependant coefficients.

1.2.4 FD-OCT or Optical Frequency-domain Imaging

Fourier transform based techniques (Also refered to as Optical Frequency-Domain

Imaging) utilize a less complex setup that requires no mechanical movement of the in-

terferometer reference arm, which provides robustness, much higher acquisition speeds

compared to time-encoded techniques and can potentially achieve much higher coherence

7

lengths and hence axial resolutions. These techniques are also refered to as Fourier do-

main OCT (FD-OCT). In such techniques, either a broadband optical source or a narrow

band, swept-tuned laser is used as a light source, spectral information of the interfero-

gram is measured and the reflection coefficient information is obtained via the Fourier

transform in wavenumber space. Fourier domain OCT can be mainly implemented in two

ways: spectral domain OCT (SDOCT), using a broadband light source and a spectrom-

eter with a multi-channel analyzer and an expensive InGaAs camera, and swept source

OCT (SSOCT), using a broadband narrow-pulse swept-laser source and a single InGaAs

photodetector.[19]

Each of these techniques have their advantages and drawbacks, however, due to the

recent advancements in swept source technology and the generally higher cost associated

with spectral analyzers and broadband optical cameras, swept source OCT related research

and implementations have thrived in the last years over SD-OCT techniques.

1.2.5 SD-OCT

In this technique, a broadband light source and an in-line optical camera are used in an

interferometric setup to consequently acheive spectral discrimination by means of Fourier

transformation, eliminating the need for mechanical movement of the reference arm, this

technique exhibits decisively faster scanning speeds and axial resolutions compared to TD-

OCT techniques, however, they exhibit a high cost due to the broadband optical sources

and detectors and a relatively more complex set up than other FD-OCT techniques (namely

SS-OCT and its variants).

1.2.6 SS-OCT

The third generation OCT technique is known as swept source optical coherence to-

mography (SS-OCT). In these technique spectral discrimination is achieved by rapidly

tuning a narrowband light source over a broad optical band-width. Since all the light is

8

Figure 1.3: Spectral-domain based OCT system. A reference mirror remains fixed, as abroadband light source is used to resolve the depth dependant coefficients.

collapsed into a single wavelength that is consequently swept across a wide range, bet-

ter signal to noise ratios, faster speeds, a better sensitivity roll-off,and tissue penetration

compared to SD-OCT is achieved [13].

Figure 1.4: Swept-source OCT system. A reference mirror remains fixed, as a "swept"optical source is used to resolve the depth dependant coefficients.

9

1.2.7 The Need For Multi-MHZ SS-OCT

Recently, there’s a growing interest in producing volumetric representations of tissue

using OCT [35], since both SD-OCT and SS-OCT setups output single lines (A-lines) that

need to be stacked together to create B-Scans, such systems are only able to produce a

3-D volumetric map (C-scan) via post-processing. First, a series of B-scan OCT images

is acquired from distinct transverse locations in the sample, the resulting sections are then

"stacked" to generate a volumetric representation of the sample’s cross section. Conse-

quently, in SS-OCT, the time required to create such 3D Volume is determined by the time

required to collect all 2D sections plus the time taken by the software to stack them to-

gether. To this end, recent progress in SS-OCT has lead to multi-MHz A-scan line rates

[34]. A 5 MHz line rate for instance allows a B-scan image of 500 lines to be created in 0.1

ms. If 500 such frames of 500 pixels in depth in the A-scan are acquired, this means a vol-

ume of 5003 of pixel data captured in approximately 0.05 s [34]. Aside from the necessary

high speed to keep acquisition time short, and the extra processing step, the acquisition of

volumetric data needs a considerable amount of storage space ( 0.8Gs/s per channel or

1.5 Gbytes per volume) to be acquired successfully. In this work, efficient algorithms and

proposed low complexity hardware implementations are presented as a solution for high

speed processing of volumetric OCT images in real time.

10

2. THE "CALIBRATION" PROBLEM

In typical swept-source OCT (SSOCT), spectral information is acquired sequentially

using a single InGaAs photodetector while rapidly sweeping the wavenumber of a narrow-

band light source over a broad optical bandwidth. However, due to some particularities of

the employed optical swept sources (refer to section 1.3) this sweeping profile might not be

perfectly linear depending on the kind of source employed. This has lead to the use of an

auxiliary device to create a reference clock used to re-sample the signal at the correct time

indexes. Usually a Mach-Zehnder interferometer (MZI) is employed as a reference path

to either create a non uniform clock that follows the phase profile of the laser by hardware

means (conventionally using the zero crossings of the signal) or to calibrate the sample

interferometer signal in post processing. This MZI reference signal can be described as

a Gaussian damped sinusoid that non-uniformly sweeps across a narrow frequency band,

this non-uniform sweep, follows the sweeping profile of the interferometer’s sample path.

When sampling the interferometric signal uniformly without the use of some re-sampling

technique, the generated images present considerable distortion, as the spectral informa-

tion is a function of wavenumber "k", not time. In order to solve this problem, calibration

needs to be performed; in this process, the acquired interferogram signal is linearized into

k-space. The process usually involves estimating the phase-frequency change profile of

the SS-OCT system via Hilbert transformation, inverse tangent and phase unwrapping.

[20] The following figure shows a representation of a typical MZI signal and it’s instanta-

neous phase profile, corresponding to the non linear sweep associated the optical source,

this profile is sometimes referred to as the "calibration profile" as it is used to interpolate

the interferometric signal at corrected k-linear instances.

11

Figure 2.1: Instantaneous phase linearization (a): unwrapped phase of the MZI (b): MZIsignal representation. Dots depict sampling indexes.

2.1 Common Calibration Techniques

Approaches to calibrate the non-uniform sweep can be further organized into two main

categories. Those that are done in post-processing (offline) and those performed in real-

time (online).

2.1.1 Spectral Phase-based Techniques

In these techniques, a curve describing the change in instantaneous frequency of the

MZI is obtained in post processing by unwrapping the instantaneous phase of a simple

Mach-Zehnder interferometer [5] (Fig. 2.1).

Knowing the physics of how an MZI signal is created, its amplitude can be expressed

as:

IMZI(t) = Cs(t) cos (∆φ(t)) (2.1)

12

Figure 2.2: Simple zero crossing detector (left) (R1 R2). Clock signal generated atzero-crossings (right).

Where C is a constant, s(t) is the spectral shape of the emitted laser beam and ∆φ is

the phase difference between the two photons added up on the reciever coupler in an

MZI. Conventionally Hilbert transformation is used to extract the instantaneous phase of

the MZI signal ∆φ [4]. Figure 2.1 shows a common calibration trace demonstrating the

instantaneous phase ∆φ of an MZI signal, an example of a k-linear sampling time stamps

is also shown.

2.1.2 Generic Real-time Calibration techniques

In the most widely accepted online calibration technique, the zero-crossings in a Mach-

Zehnder interferometer (MZI) are used to generate a non-uniform clock [6]. As these

points are determined in time sequence, they are used to index the data set of the simulta-

neously captured spectral interferogram. Thus, the OCT signal is transformed into a set of

data with identical intervals of wavenumber (k) before the Fast Fourier Transform (FFT),

this scheme is shown in Fig. 2.2. In general, software-based methods are the more pop-

ular alternative for re-sampling and typically employ interpolation to calibrate the signal

in linear k-space intervals. Usually, longer computational times are associated with these

generally more complex interpolating functions, but typically lead to better images.

13

3. CALIBRATION BASED ON A SQUARE-LAW ENVELOPE DETECTOR*

Our first proposed low complexity calibration technique, aims to provide an easy to im-

plement re-sampling scheme that would generate a clock whenever the MZI signal passes

two, predefined thresholds using a continuous time ternary encoding (CTTE) technique

[cttepaper], by implementing this non-uniform sampling clock we aim to construct a de-

vice that would linearize the interferometer in k-space while providing the system operator

with the flexibility of choosing the desired number of points per A-line, something impos-

sible to do without interpolation in common zero-crossing based re-sampling schemes, an

added advantage of the proposed method is that only one acquisition channel is needed,

effectively cutting the required storage by half. For the CTTE block to work without the

need for dynamically changing threshold levels, a constant amplitude signal is needed. In

order to remove the amplitude information of the MZI signal we implemented a very well

known demodulation scheme known as a square-law detection, which extracts the enve-

lope of an amplitude modulated signal by means of squaring and low pass filtering and

then divides the original signal by its detected envelope. This produced a signal with a

constant amplitude term.

3.1 Amplitude Correction by Means of Square-law Envelope Detection

Envelope detection is the process by which the baseband message in an amplitude

modulated signal is retrieved. For our proposed method a square law envelope detector

was used, which exhibits significantly lower complexity. We show here that the square-

law envelope detection provides results at a par with the Hilbert transformation based

demodulators for the purpose of k-domain calibration.

∗2016 IEEE. Reprinted, with permission, from A. T. Zavareh, O. Barajas, M. Serafino, J. Jo, B. Apple-gate, B. M. Sadler, and S. Hoyos, "A novel continuous time ternary encoding based SS-OCT calibration,"2016 IEEE Biomedical Circuits and Systems Conference (BioCAS), 2016.

14

An amplitude modulated signal can be described by Eq.3.2, wherem(t) is the baseband

message, Ac cos (ωct) is the carrier wave, ka is the amplitude sensitivity of the modulator

and 1 + kam(t) is the envelope of the signal. The baseband message m(t) is a low pass

signal with cut-off frequency Ω, that is M(ω) = 0 for |ω| > Ω.

IMZI(t) = Ac[1 + kam(t)] cos (ωct) (3.1)

The squarer output is then:

I2MZI(t) =1

2A2c [1 + kam(t)]2 +

1

2A2c [1 + kam(t)]2 cos(2ωct) (3.2)

The first term on the right-hand side is a low-pass signal except that the cutoff fre-

quency has been increased to 2Ω by the squaring operation. The second term has a spec-

trum centered at 2ωc.

The squared signal now needs to pass through a filter H(ω) which is a low pass filter with

cutoff frequency 2Ω, whose ideal output is then:

y(t) =1

2A2c [1 + kam(t)]2 (3.3)

Finally, taking the square root gives an output proportional to m(t) with a DC offset. No

decimation step is needed as we wish to trace the full cycle of the wave. Once the enve-

lope information has been extracted, in order to bring the amplitude of the MZI signal to a

constant level, it is necessary to divide the original MZI signal by the output of the undeci-

mated envelope detector. Our envelope detector was implemented digitally, and simulation

showed that a very low complexity 4 tap, linear phase, half band FIR filter is sufficient for

this digital demodulator to accurately estimate the signal’s envelope as shown in Fig. 3.1,

without compromising the axial resolution of the ultimately calibrated images.

15

Figure 3.1: Envelope detection process from a continuous time perspective. Figures (c) to(f) correspond to points I-IV in (b), respectively.

After the amplitude correction step, a continuous time ternary encoding scheme was

implemented to generate a k-dependent clock as illustrated in Fig. 3.2. This circuit gen-

erates a pulse whenever the amplitude of the MZI signal exceeds a predefined ternary

encoded level-sampling threshold.

3.2 Continuous Time Ternary Encoding (CTTE)

The proposed linearization method employs an encoding technique known as Contin-

uous Time Ternary Encoding (CT-TE) [5] to generate a k-domain sampling clock for the

interferometer signal. The CT-TE only fires when an amplitude variation crosses some pre-

defined thresholds. In our implementation a threshold set (Vth,L, Vth,H) was defined form-

ing an input tracking window and subsequently stored in a lookup table. Depending on the

comparison results, the sum module generates ternary state ”1” or "-1" as x(t)(CTTE in-

put) goes higher than Vth,H or lower than Vth,L, respectively; otherwise, it holds an output

of "0" unchanged.

3.2.1 CTTE Threshold Determination

By defining a parameter N, that divides the unit circle (Figure 3.2) in equidistant ro-

tational intervals, our CT-TE provides the advantage of defining the number of output

16

Figure 3.2: Continuous time level crossing circuit implementation (top): continuous timeternary encoding (CTTE) (bottom): K-linear clock generation example, with N=8 CTTElevels

values per signal cycle, giving the system designer flexibility to choose a determined out-

put points. This in turn eliminates the need for changing the MZI length path, as enough

points can be retrieved by choice of the threshold values. The non-uniform k-domain

characteristics of the simultaneously acquired interferometer signal ∆φ are effectively lin-

earized by asynchronously sampling using this non-uniform, real time clock, generated by

the proposed thereshold level technique.

3.3 Results

In order to evaluate the performance of our proposed calibration scheme simulation

was performed using SIMULINK. Performance of the proposed method in terms of image

quality was investigated by computing the axial resolution obtained over 500 realizations

17

and three different arm depths, while modifying key variables such as MZI signal ADC

resolution and N parameter choice. Fig.3.3 shows the mean axial resolution for 500 a-

lines in terms of input ADC resolution and choice of N. Axial resolution remains constant

throughout all tested ADC resolutions implying that our proposed scheme can be used to

significantly relax ADC specifications without resolution loss, which translates to an im-

mediate power saving advantage. As stated in [9] every single ADC bit reduction can cut

the power consumption by a factor of at least four. As Fig.3.3 implies the axial resolution

was constant sweeping the ADC bit from 14 to 8. That means a factor of approximately

4096 can be saved in terms of the MZI sampling ADC power consumption.

Axial resolution remains fairly constant regardless of the choice of N. However, it is

important to note that by setting N to a convenient number, the system designer can spec-

ify the number of points per cycle desired for efficient FFT calculation and b-scan recon-

struction, which constitutes an advantage for real time implementation as this is typically

performed by elongating or contracting the optical path of the Mach-Zender interferome-

ter(MZI) as described in Eq.3.5 [8],

IMZI(t) = Cs(t) cos(2πk(l2 − l1)

)(3.4)

where l1 and l2 are the interferometer arm lengths.

Finally we measured the sensitivity roll-off across seven sample arm depth indexes, as

shown in Fig.3.3. it is evident that choice of N has a negligible effect on roll-off. For our

particular setup a value of N=8 yielded the best results from those tested.

Our proposed resampling scheme shows a much lower complexity than the spectral

phase interpolation methods, that typically use a filter consisting of tens of coefficients

to realize Hilbert transformation. Recently an 18 coefficient FIR filter Hilbert transform

implementation was demonstrated [10]. The authors found the main filter block by sys-

18

Figure 3.3: Axial resolution mean value (a) to (c) different sampling arm lengths(a:514µm-b:998µm-c:1483µm), across 500 realizations as a function of ADC resolution(8bits-14bits) and N (d): sensitivity roll-off across all depths.

tematically changing the filter order while observing axial resolution, such that 18 coeffi-

cients are needed so that no artifacts such as ghosting are produced. Their axial resolution

plateaued at 8 coefficients, showing significant artifacts and image ghosting. On the other

hand, the main filtering block on our scheme is based on a much simpler 4 tap FIR filter.

As Table 3.1 suggests, our axial resolution results are tantamount to those obtained

by performing traditional spectral phase calibration methods. Those shown in the bottom

table are obtained from a theoretical Hilbert transform and not from an FIR implemented

one. In terms of image quality our results are on a par with those obtained by offline inter-

polation methods. It can be clearly observed in Fig.3.4 that for image reconstruction, the

choice of N has little effect on the overall quality of the image.

As discussed above, a significant reduction in the MZI ADC bit is achieved, the core

arithmetic is massively reduced whilst the resolution remains constant.

19

Table 3.1: Comparison between the axial resolution of the proposed method and thetheoretical spectral phase-based technique. Proposed Method (Top). Spectral phase-based (Bottom). All quantities are in µm

N Depth (µm)

514 998 14818 [Cycle−1] 11.28 11.07 11.38

12 [Cycle−1] 11.74 11.72 11.6816 [Cycle−1] 11.09 11.49 11.38

Quasi N Depth (µm)

514 998 14818 [Cycle−1] 11.10 11.10 11.09

12 [Cycle−1] 11.11 11.11 11.1416 [Cycle−1] 11.13 11.12 11.12

Figure 3.4: Image reconstruction comparison. (a) Uncalibrated. (b) through (d), increasingnumber of angular quantization levels N.

20

3.4 Conclusion

In this section, we have leveraged a novel, computationally efficient, less power con-

suming approach for spectral calibration of swept sources using a square law envelope

detector and a continuous time ternary encoder (CT-TE) sampler. A corrected k-linearized

clock was extracted from a simple Mach-Zehnder interferometer (MZI) and then used to

re-sample the interferometer signal at the corrected time indexes. Axial resolution showed

results on par with a conventional spectral phase-based calibration technique while main-

taining a significantly reduced computational complexity enabling for online implementa-

tion of the method.

21

4. KALMAN FILTERING BASED CALIBRATION*

Although faster algorithms for simple calibration have been presented [14], none of

these algorithms account for the presence of noise in the input signal. This can limit their

application in ultra low cost, custom OCT systems. The main motivation for this work

is to provide a framework that facilitates less complex implementation of cost effective

swept sources and ultimately lead to a wider adoption of the technology.

4.1 Setup

A custom-built SS-OCT system (Fig. 4.1) was used for acquisition; it is comprised of a

broadband Swept source (ESS, Exalos) with a center wavelength of 1310 nm and 150-kHz

sweep rate. The signals were aquired using a 14-Bit 250 MS/s digitizer Module (NI5761,

National Instruments).

As shown in Fig. 4.2, the digitally acquired MZI signal data is fed into an Extended

Kalman Filter algorithm block that estimates the instantaneous phase, which is then pro-

cessed by the comparison step block (Fig. 4.4) to create an asynchronous K-linear clock.

This CLK signal is then used to acquire the interferometric signal at the correct time in-

stants.

4.1.1 Estimation

For the estimation of the MZI signal instantaneous frequency φ an Extended Kalman

Filter (EKF) was used (Fig. 4.2). An Extended Kalman Filter is generally used to estimate

the values of inner state variables of a non-linear system model with known outputs and

noise statistics [15]. This poses multiple implementation challenges due to some pecu-

∗2017 IEEE. Reprinted, with permission, from A. T. Zavareh, O. Barajas and S. Hoyos, "An EfficientEstimitaion Algorithm for the Calibration of Low-Cost SS-OCT Systems," 2017 IEEE 14th InternationalSymposium on Biomedical Imaging (ISBI), Melbourne, Australia, 2017.

22

Figure 4.1: SS-OCT optic elements configuration. PC: polarization controller, ODL: opticdelay line.

Figure 4.2: Estimation technique block diagram. σ2w is the measurement noise power. σ2

nA

is the amplitude noise power. σ2nφ

is the instantaneous phase noise power

liarities of the MZI signal such as the chirp not being constant in amplitude, the instan-

taneous phase not increasing monotonically, and the phase being highly noisy. However,

all of these were overcome by generating an accurate state space model as subsequently

described.

State Space modeling: A nonlinear state space model can be written as:

X[n] = f(X[n− 1],N[n− 1]),

Y[n] = h(X[n],W[n]),

(4.1)

where X is the matrix of state variables, N is the state noise, Y is the output measurements,

and W is the measurement noise.

23

4.1.1.1 State equations

The state variables are constructed as follows [15],

X[n] =[A[n] φ[n] φ(1)[n] φ(2)[n] φ(3)[n]

], (4.2)

where φ is the instantaneous phase, φ(n)[n] is the nth order derivative of φ, and A is the

instantaneous amplitude.

A sampled MZI can be modeled as [15],

y[n] = A[n]cos(φ[n]) + W[n] = A[n]cos(M∑i=l

bini) + W[n], (4.3)

where bi are fixed polynomial coefficients. The assumption is that the instantaneous phase

profile is ruled by an M th order polynomial behavior.

Using discrete time Taylor expansion, the state update equation can be written as,

X[n] = FX[n-1] + N[n-1] =

1 0 0 0 · · · 0

0 1 11!

12!· · · 1

M !

0 0 1 11!· · · 1

(M−1)!

......

...... . . . ...

0 0 0 0 0 1

X[n− 1] +

nA[n− 1]

nφ[n− 1]

nφ(1) [n− 1]

...

nφ(M) [n− 1]

. (4.4)

In the above equation, the amplitude of the MZI signal has been modeled as what is

known as a random walk in literature [16]. This model assumes that the amplitude in

time n differs from the amplitude in time n-1 by a Gaussian, stationary white amplitude

noise nA[n]. It can be observed that the phase profile of a typical MZI exhibits a third

order polynomial behavior with its second order derivative resembling a random noise

24

Figure 4.3: MZI signal parameter estimation for a typical realization. (a) Amplitude pa-rameter extraction. (b) Instantaneous phase extraction

due to intrinsic measurement error. This poses a problem as the filter can estimate noise

as a constant unless accounted for. Consequently then, any phase profile that exhibits a

higher order than two cannot be evaluated. To account for this, we introduce a phase noise

parameter nφ[n] in Eqn. (4.4) that allows for higher order correlated product determination

as shown in Eqn. (4.5).

By accounting for noise in the system, we effectively give the swept source designer

the ability to use lower quality photo electric components, significantly reducing the cost

of implementation and allowing for a wider adoption of SS-OCT systems.

It can be easily presumed that nφ(l) [n] = nφ(l−1) [n]− nφ(l−1) [n− 1]. Subsequently, the

state noise covariance matrix V, can be written as,

V = Cov(N) =

σ2nA

0 0 0 0

0 σ2nφ σ2

nφ σ2nφ σ2

nφ

0 σ2nφ 2σ2

nφ 3σ2nφ 4σ2

nφ

0 σ2nφ 3σ2

nφ 6σ2nφ 10σ2

nφ

0 σ2nφ 4σ2

nφ 10σ2nφ 20σ2

nφ

(4.5)

25

Figure 4.4: Comparison step circuit. (Top left) Equally divided calibration profile. ∆defines the number of points designed to be sampled (wave number quantization interval).(Bottom Left) Calibrating CLK. (Right) Asynchronous time comparison block diagram.

4.1.1.2 Observation equations

Since the parameters this work is trying to estimate are not directly measurable, a

nonlinear Extended Kalman Filter model was used to relate state variables to the measured

data. Simplifying Eqn.(4.1) we have,

Y[n] = h(X[n],W[n]) = x1[n]cos(x2[n]) + W[n], (4.6)

where W[n] is modeled to be a stationary, white, Gaussian noise with EW[n]W[n+

k] = σ2w

2δ[k].

This nonlinear equation must be linearized using the Jacobian matrix. That is

H[n] = [hij]1×(M+2) =∂hi∂xj

∣∣∣∣x−n

, (4.7)

where x−n is a priori state estimate at step n given knowledge of the process prior to step n.

That means H[n] can be written as,[cos(x−2 [n]) −x−1 [n]sin(x−2 [n]) 0 0 0

].

Once the model is complete and the equations derived, the proposed implementation

26

for the presented amplitude and instantaneous phase estimation method is shown in Al-

gorithm 1. The addition of amplitude, phase and measurement noise into the state space

model improves robustness to noisy input signals. Figure 4.3 shows the instantaneous

phase and amplitude profile for one particular MZI signal realization, as estimated by the

filter. Although both the amplitude and phase content can be used for calibration, the in-

stantaneous phase was preferred in this work because it enables a simpler implementation

of the asynchronous clock generation circuit.

4.1.2 CLK Generating Mechanism

The light intensity profile of the MZI output can be described by,

IMZI(t) = CS(t) cos(φ(t)), (4.8)

where C is a constant, S(t) is the spectral shape of the emitted laser beam and φ is the

phase difference between the two photons added up on the receiver coupler in an MZI.

The extracted instantaneous phase, φ, is driving the comparator to generate a calibrat-

ing asynchronous clock (Fig. 4.4) [17]. The comparison step only fires up a positive edge

when the phase is increased by a predefined value (∆). This method will effectively gener-

ate positive edges of a clock signal whenever the phase is incremented by equal intervals.

[h] Extended Kalman Filter [11]

Initialization:

P0 = I (Identity matrix). x0 = Initial estimate of the state space model. n = 1

Time update:

x−n = Fxn−1 P−

n = FPn−1FT + V Measurement Update:

Kn = P−nHT

n

(HnP−

nHTn + σ2

w

2

)−1

. xn = x−n + Kn(yn −Hn). Pn = (I − KnHn)P−

n .

close;

27

Figure 4.5: Image reconstruction comparison. (a) Uncalibrated. (b) through (d), increasingnumber of wavenumber quantization levels P.

4.2 Results

Performance of the presented algorithm was evaluated in terms of axial resolution,

sensitivity roll-off and image reconstruction. Data for over 500 realizations was obtained

for three different arm depths while modifying the P parameter (number of sampling points

per MZI cycle); we also compared the performance of the algorithm against different ADC

resolutions. Fig. 4.6 shows the mean axial resolution for 500 a-lines in terms of input

ADC resolution and sample point number. Axial resolution remains constant throughout

all tested ADC resolutions, implying that our proposed scheme can be used to significantly

relax ADC specifications without resolution loss, which translates to an immediate power

saving advantage. A single bit reduction can represent a power consumption reduction

factor of at least four [14].

Finally we measured the sensitivity roll-off across seven sample arm depth indexes,

as shown in Fig. 4.6. Sensitivity roll-off remained acceptably low throughout the entire

28

Figure 4.6: Axial Resolution mean value (a) to (c) Different sampling arm lengths(a:1448µm-b:1921µm-c:2361µm), across 500 realizations as a function of ADC Reso-lution (8bits-14bits) and P (d): sensitivity roll-off (20log10, dB) across all depths.

imaging depth, without showing significant decay until after 1.5mm.

Our proposed resampling scheme exhibits results comparable to offline calibration

methods while providing robustness to noise and efficiency, making it suitable for real-

time, low cost SS-OCT implementations. While our in-house system was built for $30000,

the state of the art commercial products can cost as high as $70000 (THORLABS, Part

number : TEL1300V2).

In terms of image quality our results are on a par with those obtained by offline cali-

bration methods. It can be clearly observed in Fig. 4.5 that for image reconstruction, the

choice of P has little effect on the overall quality of the image.

4.3 Conclusion

We demonstrated a novel approach for the spectral calibration of low cost, noise prone,

swept sources using a real-time estimation algorithm and a very simple sampling scheme

29

that allows for effortless determination of the number of sampling points needed in an

B-scan. A corrected k-linearized clock is estimated from a simple Mach-Zehnder inter-

ferometer (MZI) and then used to non-uniformely sample the interferometer signal at the

correct times. Axial resolution results were found to be on par with conventional spec-

tral phase-based calibration techniques while significantly improving robustness, enabling

future development of cost effective SS-OCT systems.

30

5. INTERPOLATED DISCRETE FOURIER TRANSFORM BASED

CALIBRATION*

Instantaneous frequency and phase estimation of time varying signals has been inves-

tigated thoroughly in multidisciplinary applications including motor fault detection and

RF demodulation. By pioneering in comparing various known IpDFT frequency estima-

tion algorithms against the state of the art, this work aims to provide a comprehensive

insight into on chip solution implementation for common SS-OCT processing techniques

including calibration[31], dispersion correction, and background subtraction. Repeated

application of IpDFT on buffered pieces of the Mach-Zehnder Interferometer (MZI) signal

was first performed in this work to generate spectral frequency domain calibrating clock.

As will be discussed further in the following sections, the significant inherent advantages

of the proposed technique make it conclusively suitable for silicon based calibration solu-

tions. Our simulations show a significant improvement in the execution time, dynamic and

static memory for the recommended technique are both down due to less computation run-

time and less complexity, respectively. The auxiliary circuitry needed to employ IpDFT is

also much less convoluted than those of the Hilbert transformation as the phase extraction

functions for the state of the art is reported to be extremely costly to be implemented. [21]

5.1 Setup

Data collection was implemented in a purposely built SS-OCT system comprised of

a swept source with a center wavelength of 1310 nm at 150 kHz sweep rate (ESS, Ex-

alos). Acquisition was performed using a 14-Bit 250 MS/s digitizer (NI5761, National

Instruments).∗2017 IEEE. Reprinted, with permission, from O. Barajas, A. Tofighi Zavareh, and S. Hoyos, "High

Performance Calibration Algorithm for Ultra High Speed Swept Source Optical Coherent Tomography",2017 IEEE International Symposium on Circuits and Systems (ISCAS), 2017.

31

Figure 5.1: IpDFT algorithm implementation diagram, bottom right shows a split-radixFFT butterfly diagram.

5.2 IpDFT Implementation

Interpolated DFT (IpDFT) [21] is an established discrete Fourier transform based tech-

nique for the accurate calculation of the instantaneous parameters of a signal. IpDFT meth-

ods improve upon the fundamental problems exhibited by classical DFT algorithms such

as spectral leakage, and can provide a faithful frequency estimation on small observation

windows. In this work, the applicability of such algorithms for the calibration problem

was thoroughly evaluated and implemented as follows: The MZI signal is acquired and

buffered in blocks of size P, these get multiplied by an appropriate time window and lastly,

an efficient split radix FFT [22] algorithm calculates the current block DFT bins so that a

correct instantaneous frequency can be obtained via one of the proposed techniques in this

section (Fig. 1).

32

Table 5.1: Computational complexity comparison of a Hilbert transformation based tech-nique against IpDFT utilizing multiple windows and enhanced split radix FFT. Signallength is defined as N.

Techniques Hilbert IpDFT [21]

Order L Obs Window Length P

# Add / per output L P(43log2(P)-8

9)-1

9(-1)log2(P )

# Mult / per output L P(23log2(P)-19

9)+1

9(-1)log2(P )

# Total Arithmetic 2LN NP

(2Plog2P-3P)

5.2.1 Time Windows

Time window properties can be evaluated by their main lobe width and side lobe atten-

uation. While the former is designed to be narrower to increase the estimation resolution,

the latter tends to be as big as possible to minimize the spectral leakage.

These trade-offs should be taken into account when the length of a window is chosen.

Higher order Rife-Vincent Class I (RVCI) windows may result in negligible estimation

error because of the fast side lobe decay. Still, the signal needs to contain enough num-

ber of periods due to wide main lobe of such windows. Although the systematic errors

for Kaiser-Bessel and Dolph-Chebyshev windows are significantly higher than high order

RVCI windows due to lower side lobe decay rate, they can provide a better frequency es-

timation for signals with 2-4 cycles. It should be noted that the use of time windows can

be eliminated in some IpDFT techniques such as Bertocco-Yoshida.

5.2.1.1 Background Theory

The MZI signal at any buffered block is modeled by,

y[n] = Acos(ω0n+ φ)e−dn =

A

2

(e−dnej(ω0n+φ) + e−dne−j(ω0n+φ)

),

(5.1)

33

Table 5.2: Multiplication algorithms complexity, log∗ stands for iterated logarithm.Techniques Multiplication

Order N

Furer’s algorithm [23] O(Nlog(N) 2O(log∗(N)))Schonhage Strassen algorithm [24] O(Nlog(N)log(log(N)))

Mixed level Toom-Cook [25] O(N 2√

2log(N)log(N))

Table 5.3: Comparison between the complexity of auxiliary blocks needed for Hilberttransformation techniques and IpDFT. M(N) below stands in for the complexity of thechosen multiplication algorithm.

Hilbert IpDFT

Tangent Inverse Max Bin Det

Taylor series; FFT-based acceleration [26] O(M(N)N1/3 log2(N)) Compare in Pairs O(P )Arithmetic-geometric mean iteration [27] O(M(N)log(N)) Tournament Method O(P )

Taylor series; binary splitting + bit burst [28] O(M(N)log2(N)) Linear search O(P )

where d is the damping factor and ω0 is the center frequency. It can be shown that the DFT

of the first term in the right hand side of Eqn. (5.1) can be written as,

DFTe−dnej(ω0n+φ) = ejφ1− e(jω0−jωk−d)P

1− ejω0−jωk−d, (5.2)

and the entire DFT can be approximated as,

Yk ≈A

2

(ejφ

1− λP

1− λe−jωk

), 0 ≤ ω < π, (5.3)

where λ = e−d+jω0 . Y ∗k is the maximum height bin in the corresponding DFT vector. k

denotes the bin index at which the maximum bin lands.

34

5.2.1.2 Betrocco-Yoshida (BY-2) algorithm

Let us assume that the adjacent bins to Y ∗k are ordered as Y ∗

k−1 being the highest fol-

lowed by , Y ∗k+1, and Y ∗

k−2. The value of λ can be proven to be,

λ = ejωk1−R

e−j2π/P −Re−j2(2π/P ), (5.4)

where,

R =Y ∗k−2 − 2Y ∗

k−1 + Y ∗k

Y ∗k−1 − 2Y ∗

k + Y ∗k+1

. (5.5)

In the case where the highest adjacent bin is Y ∗k+1 followed by Y ∗

k−1 and Y ∗k+2,

λ = ejωk1−R

e−j2(2π/P ) −Re−j2π/P, (5.6)

where,

R =Y ∗k−1 − 2Y ∗

k + Y ∗k+1

Y ∗k − 2Y ∗

k+1 + Y ∗k+2

. (5.7)

Following the discussion above, ω0 = ImLn(λ), and d = ReLn(λ).

5.2.1.3 RVCI windows (Order M)

IpDFT algorithms can be analytically derived for RVCI windows. In these algorithms

a RVCI window is multiplied to each chunk of the data the instantaneous parameters are

calculated for,

y[n] = wnxn = wnAcos(ω0n+ φ)e−dn = wnAcos(ω0n+ φ). (5.8)

35

After taking the DFT from the buffered signals, the parameters R1 and R2 are defined as,

R1 =‖Y ∗

k+1‖2

‖Y ∗k ‖2

≈ (δ +M)2 +D2

(δ −M − 1)2 +D2,

R2 =‖Y ∗

k−1‖2

‖Y ∗k ‖2

≈ (δ −M)2 +D2

(δ +M + 1)2 +D2,

(5.9)

where D = dP /(2π). Computing the mentioned ratios, δ can be calculated as,

δ = −2M + 1

2

R1 −R2

2(M + 1)R1R2 −R1 −R2− 2M. (5.10)

From (5.9) the damping factor is,

d =2π

P

√(δ +M)2 −R1(δ −M − 1)2

R1 − 1, δ > 0,

d =2π

P

√(δ −M)2 −R2(δ +M + 1)2

R2 − 1, δ < 0.

(5.11)

The parameter ω0 then is going to be equal to (k + δ)2πP

.

It should be noted that RVCI window order zero is called a rectangular window and the

same kind with an order two is called Hanning window. [21]

5.2.1.4 Clock generation

Knowing the instantaneous frequency of each block of size P, a sampling clock was

generated by creating a piecewise linear signal with the slope of each line being propor-

tional to the next transitory frequency of the MZI. Previously proposed continuous-time

level sampling techniques in [29] are used to collect the correct sampling time stamps.

36

5.2.2 Algorithmic Complexity Comparison

This section compares the complexity of state of the art Hilbert transformation tech-

niques with IpDFT based techniques in terms of core arithmetic complexity and auxiliary

operations needed for calibration.

5.2.2.1 Core Operation Time/Arithmetic Complexity

Until now, the most efficient reported techniques for the spectral calibration of SS-

OCT systems involved the use of FIR Hilbert transformers as a core block. For the IpDFT

algorithms, the main block consists of a split radix DFT block with a signal length of P

performed on blocks of the MZI signal.

As Table 5.1 shows, an FIR filter with an order L needs L additions and L multipli-

cations for each output sample to be executed. For a MZI signal of length N that implies

a total arithmetic operations count of 2LN. To perform the DFTs in IpDFT, we used an

efficient split-radix algorithm. The complexity and the number of additions and multipli-

cations needed to perform the algorithm is shown in Table 5.1.

In the case of our application the signal length N was 3584 and the observation win-

dow length P, was chosen to be 64 samples. The state of the art Hilbert transformation

technique in literature [30] uses 17 FIR filter coefficients to implement Hilbert transfor-

mation. Simple arithmetic calculations yield that Hilbert transformation based technique

needs 121856 operations to compute the analytic signal while the proposed method only

uses 32256 operations. It should be stated that a split radix needs only log2(P ) clock cy-

cles to be computed for each instantaneous phase sample, while Hilbert transform takes L

clock cycles.

37

5.2.2.2 Auxiliary Operations Time/Arithmetic Complexity

In order to obtain the instantaneous phase using Hilbert transformation, an additional

inverse tangent step needs to be performed on the analytical signal. It is worth noting

that inverse tangent is a particularly difficult and inefficient operation to implement. De-

termination of inverse tangent algorithm complexity can itself be intricate, as the overall

number of operations is a function of the chosen multiplication type; to give the reader

a sense of the overall inverse tangent complexity, some notable multiplication algorithms

are included in Table. 5.2, while Table. 5.3 contains applicable algorithms for inverse

tangent determination. For IpDFT however, the only extra auxiliary operation needed is

maximum DFT bin determination. Table 5.3 clearly shows that performing inverse tangent

is significantly more complex than the techniques used to find the maximum DFT bin.

Figure 5.2: (a) Normalized execution time compared to Hilbert transformation vs. Ob-servation window length, (b) Axial resolution compared to sample arm depth, (c) Axialresolution vs. Observation window length.

38

5.3 Results

The proposed algorithm performance was evaluated in terms of axial resolution, exe-

cution time, and image quality. Figure 5.2(a) shows the normalized execution time for 500

interferometer realizations compared to Hilbert transformation. It can be easily observed

that Betrocco-Yoshida (BY-2) algorithm is by far the most efficient, outperforming Hilbert

for up to a factor of ten depending of chosen P factor. The logarithmic behaviour predicted

in Table 5.1 can also be observed from Figure 5.2.

To evaluate the axial resolution, 500 A-scans were acquired from a mirror sample at 5

different arm depths for all three different IpDFT methods. As shown in Figure 5.2(b),

the Yoshida algorithm also exhibits the best result in terms of axial resolution up to an

observation window length of 64.

As shown in Figure 5.2(c), observation window length was swept across all proposed

calibration methods in order to determine its optimum size; generally, an ascending trend

in axial resolution was observed as window length increased. It can also be observed that

for IpDFT1 and IpDFT3, at least 64 samples are needed for calibration to be performed.

The designer needs to always keep in mind that extremely short windows can not spot the

temporal specifications of the MZI.

Figure 5.3 shows the cross sectional image of a biological tissue calibrated by different

techniques. Buffer size was chosen to be 64 in order to accurately compare all methods.

5.4 Conclusion

We demonstrated the superiority of Interpolated Discrete Fourier Transform (IpDFT)

based algorithms over state of the art FIR Hilbert calibration techniques for the spectral

calibration of SS-OCT systems. Bertocco-Yoshida (BY-2) decisively exhibited improved

results in terms of efficiency, complexity and axial resolution. More importantly, we laid

foundational work for future low cost, system on chip solutions for SS-OCT calibration

39

and other necessary signal conditioning steps, eventually leading to substantial cost and

power advantages.

Figure 5.3: Image reconstruction comparison. (a) Uncalibrated. (b) Bertocco-Yoshida(BY-2) (c) RVCI (M=1) (d) RVCI (M=3)

40

6. SUMMARY AND CONCLUSIONS

Swept Source Optical Coherence Tomography has recently become the de-facto imag-

ing technique when high axial resolution is needed in low penetration applications. With

the recent advancements on swept source technology and volumetric microscopy imaging,

new challenges in terms of processing and storage of large datasets have emerged. In this

thesis, a multitude of spectral calibration algorithms were presented, all attempting to face

one of the main identified challenges surrounding SS-OCT technology: Ultra-Fast pro-

cessing for MHz SS-OCT, Low data throughput for fast and efficient C-Scan generation

and noise tolerance for novel cost effective swept source implementations like MEMS-

less VCELS. Promising results were exhibited by all the techniques in their respective

application.

6.1 Challenges

A comprehensive study needs to be performed to identify the key challenges and op-

portunities regarding the development of efficient signal processing of solutions for SS-

OCT systems, preliminary research indicates such algorithms may contribute to the devel-

opment of low cost, all inclusive solutions for the spectral calibration of SS-OCT systems.

6.2 Further Study

A hardware implementation is currently being validated on an FPGA platfrom to in-

vestigate the real world application of the techniques developed in this work.

41

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